An objective method for determining principal time
scales of coherent eddy structures using
orthonormal wavelets
Jozsef Szilagyi
a,*, Marc B. Parlange
b, Gabriel G. Katul
c& John D. Albertson
d aConservation and Survey Division, Institute of Agriculture and Natural Resources, University of Nebraska-Lincoln, 113 Nebraska Hall, Lincoln, NE 68588-0517, USA
b
Department of Geography and Environmental Engineering, The Johns Hopkins University, Baltimore, MD, 21218-2686, USA c
School of the Environment, Duke University, Durham, NC 27708-0328, USA d
Department of Environmental Sciences, University of Virginia, Charlottesville, VA, 22903, USA (Received 13 April 1998; accepted 28 October 1998)
A new, parameter-free method, based on orthonormal wavelet expansions is proposed for calculating the principal time scale of coherent structures in atmo-spheric surface layer measurements. These organized events play an important role in the exchange of heat, mass, and momentum between the land and the atmosphere. This global technique decomposes the energy contribution at each scale into organized and random eddy motion. The method is demonstrated on vertical wind velocity measurements above bare and vegetated surfaces. It is found to give nearly identical results to a local thresholding approach developed for signal de-noising that assigns the wavelet coecients to organized and random motion. The eect of applying anti- and/or near-symmetrical wavelet basis functions is also investigated. Ó1999 Elsevier Science Limited. All rights
re-served
1 INTRODUCTION
The important role of coherent structures in the atmo-spheric boundary layer (ABL) for the exchange of heat, mass, and momentum across the land-atmosphere in-terface is well established4,16±18,27,38,40,44. According to Wilczak's44 de®nition, coherent structures are distinct large-scale ¯uctuation patterns regularly observed in turbulent ¯ows. These patterns or regions of the ¯ow ®eld are spatially and temporally correlated with them-selves and statistically signi®cant with respect to the turbulence energy of the ¯ow41. When viewed in this way, turbulent ¯ows are comprised of a superposition of organized, coherent structures and disorganized, or ``random'' background activity21,43.
Coherent structures in the atmosphere can be so markedly developed that they are clearly identi®able in measured time series as ``ramps'' arising from the
sweep-ejection motion in the surface layer1,16,17,19,27,36,43. Gao
et al.,16,17 identi®ed coherent structures above a forest canopy having regularly repeated organized cycles of ejection-like upward ¯ow and the subsequent sweep-like descending motions. Shaw et al.,40 identi®ed coherent bursting structures in turbulent shear ¯ows for plant canopies. A large amount (>75%) of the total vertical momentum and heat ¯ux was found to take place in these bursting events.
Wavelet transforms have become an important tool for identifying and studying coherent struc-tures7,13,15,19,25,26,30±32,43. These wavelet-based tech-niques often provide useful information which is advancing our understanding of turbulent transport, for example, the ¯ux of water vapor into the atmo-sphere21,23,24.
Wavelet transforms allow the decomposition of data into dierent frequency or scale components distributed in space8,28±30,33,34. Although there are many types of wavelet transforms, they can all be classi®ed as either continuous or discrete. In the present study discrete Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter
PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 4 6 - 3
*
Corresponding author. E-mail: jszilagy@unlinfo.unl.edu
orthonormal wavelet expansions are applied to turbu-lence measurements in the atmospheric surface layer to identify the principal time scales of the coherent struc-tures. As suggested by Yamada and Ohkitani45, ortho-normal wavelet transforms may be preferable since the orthogonality minimizes the number of wavelet coe-cients and suppresses undesired relations between the coecients.
For a discrete one-dimensional signal f(j), discrete orthonormal wavelet coecients are given by,
w mi X
N
j1
g miÿ2mjfj; 1 where w m[i] is the wavelet coecient at locationi and
scalem, andNis the total number of data points, which must be an integer power of two. The variableg m(i) is
the dilated version of the discrete orthonormal basis function g 0such that6,8,28,29
g mi 2ÿm=2
g 0 i 2m
: 2
Note that in this arrangement, at scale indexm1 there are N/2 wavelet coecients, at scale index m2 there are N/4 coecients, and so on, up to the single coe-cient at scale index mM, where Mlog2N. The
original signalf(j)can be reconstructed from its wavelet coecients using
In this paper three anti-symmetric (Daubechies-4, Daubechies-6, Daubechies-8) and three near-symmetric (Symlet-8, Symlet-12 and Symlet-16) basis functions (wavelets) are employed. The number in the name of the wavelet refers to the number of coecients necessary to de®ne the speci®c wavelet. This is done in order to study the eect of the choice of basis function on the analysis. Note that entirely symmetric orthonormal wavelets do not exist9. Consequently we applied the least anti-sym-metric basis functions available9.
Coherent structures or organized events in atmo-spheric time series are typically obscured by the super-imposed incoherent part of the signal19. Wavelet decomposition allows us to separate the scales while maintaining the temporal reference. Regions of the time-scale half plane with high magnitudes of the squared wavelet coecients correspond to strongly developed organized events at the given scale and time. As an ex-ample, the wavelet decomposition of a velocity signal measured in the lower atmosphere is presented in Fig. 1. Note that, near the time 260 s, and scale 128 s a strong coherent structure in the velocity signal is indicated by high values of the squared wavelet coecients. The square of each wavelet coecient, when scaled appro-priately, is proportional to the contribution of the actual structure at time i and scale m to the variance of the measured signal. This is so because the following holds
(Parseval's identity) for the scaled wavelet coecients
w^ m i
Most methods proposed to identify coherent structure properties have subjective compo-nents2,3,5,14,20,27,36,37,39,40. For example, an arbitrary threshold value is set for the wavelet coecients when selecting organized events, or the size of a certain win-dow employed in the analysis must be speci®ed. The result of the analysis, therefore, may depend on arbi-trarily de®ned parameters that do not translate from one experimental site to the next.
In this study an objective, so-called global, method is proposed for calculating the principal duration time of coherent structures based upon discrete orthonormal basis functions. When applying the technique, the principal time scale is de®ned to be twice the scale in the time-scale half plane where the dierence between the normalized (i.e. sum of the squared wavelet coecients at the given scale divided by the sum of the variances of all scales) energy content of the measured data and that of a white noise process is maximum. The premise is that the larger the dierence for each scale the more the data dier from a white noise process at those scales and consequently, the more intense is the role of the coherent structures at the corresponding scales.
The calculated principal time scales are compared with the results of a local technique proposed by Don-oho and Johnstone10,11for signal de-noising. There only those wavelet coecients are kept for the energy content calculation of dierent scales that are larger in magni-tude than a certain universal threshold value. Since the low-magnitude wavelet coecients, which are supposed to represent disorganized, random motion (i.e. noise) of the ¯ow are discarded (set equal to zero), the remaining Fig. 1.A sample 1s block averaged longitudinal velocity time series and its discrete orthonormal wavelet transform with contours of the squared wavelet coecients. The
wavelet coecients are implied to be connected to the organized events. The principal time scale in this case is selected as twice the scale at which the normalized variances (sum of the thresholded squared wavelet co-ecients) display a maximum.
2 DATA COLLECTION
Atmospheric surface layer measurements were carried out at the University of California, Davis, Campbell Tract Research Facility over a bare soil ®eld (24 June, 1994 from 9:20 am to 3:00 pm) and over a bean crop ®eld (29 August, 1994 from 9:20 am to 1:40 pm). The bare soil ®eld had an average momentum roughness height (z0) of 2 mm, while the bean ®eld had an average
momentum roughness height of 30 mm at the time of measurement. The vertical component of the wind ve-locity was recorded at 21 Hz by an ultrasonic ane-mometer located at 1.5 m above the soil surface and written into ®les of 20 min each.
The analysis is based on a total of 17 twenty-minute ®les for the bare soil and 13 for the bean ®eld. Throughout the measurement periods unstable atmo-spheric strati®cation conditions prevailed.
Prior to analysis, the 21 Hz measurements were block averaged to 1 Hz, since the principal time scales of the coherent structures are expected to be at least an order of magnitude greater than 1 second19,36. As a result, the original 20 min data sets were reduced to 1024 (210)
data points.
3 ALGORITHMS
3.1 Proposed global method
The more distinct a turbulent structure is the higher will be the value of the squared wavelet coecient at the corresponding location and scale33. Consequently, it is possible to estimate the relative contribution of the dierent coherent structure scales to the total variance of the signal using
wherepmis the portion of the total variance of the signal (see (4)) that can be attributed to structures with dura-tion time d equal to 2m1 (s). Note that the structure
duration time d is de®ned as twice the actual wavelet scale after Lu and Fitzjarrald27, who argue that the wavelet scale is comparable to the time period of up-drafts for each coherent structure. The smallest structure detectable in this study has a duration of 4 s, since the Nyquist frequency for the block averaged data is 0.5 Hz.
As one might expect, the pm values (m1,..M) range between zero and one, and sum to unity.
If the data represent a white noise (wn) process, the expectation of the pm values can be expressed as
hpwnm d 2m1i
where we make use of the white noise property that the expected value of the squared and properly scaled wavelet coecients is constant for each scale.
The maximum of thepmvalues for (5) does not signify the predominance of the dierent structure durations, since even in the case of a white noise process the ex-pectation of the pm values are not evenly distributed across the scales (see Fig. 2). This is because the number of wavelet coecients to be summed for each scale de-creases exponentially with increasingm, resulting in the general exponential type decay of the pwn
m values dis-played in Fig. 2. In order to obtain a useful estimate of the relative importance of the dierent coherent structure scales, the pwn
m values for each scale of the white noise process have to be subtracted from the samepmvalues of the turbulent measurements. This way, the remaining part of the pm values are not the result of disorganized, random motions. The greater this dierence for a given scale, the more organized is the measured signal at that particular scale. In this way, the scale with the largest positive value in the dierence (pmÿpwn
m ) corresponds to (one-half) the principal time scale of the coherent struc-tures in the turbulent ¯ow. In Fig. 3(a), (b) these dier-ences are presented applying anti- and near-symmetric wavelets for the vertical component of atmospheric ¯ow measurements over bare soil and a bean ®eld,
respec-Fig. 2.Normalized energies (pm) of the wavelet scales. White
tively. The peak in each graph marks the principal time scale of the coherent structures.
3.2 Donoho and Johnstone's local method
Donoho and Johnstone10,11 and Donoho12 proposed a universal thresholding value for noise separation of multidimensional signals when wavelet transforms are applied. The underlying idea is that the noise of the signal is represented by mostly low-magnitude coe-cients in the time-scale half plane. Using only the high-magnitude wavelet coecients in the inverse transform, the original signal can be reconstructed with dramatic reduction in the noise level11. We assume that the same approach can be used for the detection of coherent structures in a turbulent ¯ow, if the disorganized part of the ¯ow is considered as a noise. Donoho and John-stone10,11showed that choosing a threshold value, th as
thr 2ln K1=2; 7 where K is the number of wavelet coecients at the ®nest scale (512 in our case), and r is the estimated median absolute deviation (divided by 0.6745) of the wavelet coecients at the same scale, provides near-optimal performance in noise reduction if the noise is white. After comparing the magnitude of each wavelet coecient with the value of th and setting all coecients equal to zero that are smaller in magnitude than th, (5) was employed. The resulting pth
m values for the vertical velocities recorded over bare soil and the bean ®eld are displayed in Fig. 3(c), (d).
Comparing Fig. 3(a), (b) with Fig. 3(c), (d), it can be seen that the two methods (i.e. global and local) resulted in a principal time scale of 16 s for the vertical velocities over both ®elds. Fig. 3(c), (d) displays wider error bars than Fig. 3(a), (b) due to the smaller number of the wavelet coecients employed in the local method. Fig. 3.Normalized and corrected variances (pmÿpwnm ) of the wavelet scales (mean and standard error) by the global method using
anti- and near-symmetric wavelets. Also normalized thresholded variances (pth
m) of the wavelet scales (mean and standard error) by
4 DISCUSSION
Based upon the above described methods of identifying coherent structures, the analysis of vertical velocities over bare soil and a bean ®eld revealed the following:
(1) The principal time scale of coherent structures for the vertical velocities was found to be around 16 s over both ®elds (bare soil and bean). This is in agreement with Qiuet al.,36 who applied pseudo-wavelet analysis for coherent structure identi®cation for ¯ow over a maize ®eld under similar meteorological conditions. They found the same principal time scale for the vertical velocities (about 15 s) at the top of the canopy.
(2) The resulting principal time scales were only slightly dependent (due to minor scatter of points in Fig. 3(a)±(d)) on the choice of the basis function within the same type (i.e. near-symmetric or anti-symmetric). The two methods (i.e. global. vs. local) gave identical results with the global method expressing less scatter due to the larger number of wavelet coecients employed in the technique. Since Hagelberg and Gamage22 demon-strated that anti-symmetric wavelets are more sensitive in detecting zones of sharp transitions connected to coherent structures, the use of anti-symmetric basis functions is perhaps better suited to the present eort.
In summary, the application of the global method, with the use of anti-symmetric wavelets, is recom-mended for coherent structure principal time scale cal-culations because the technique retains all the wavelet coecients fully representing the original signal. The principal time scale in the method is de®ned as twice the scale where the associated energy (calculated via the squared wavelet coecients) has a maximum above a reference (white noise) energy level. In contrast, thresholding methods discard squared wavelet coe-cients with values smaller than a certain threshold value before calculating energies distributed over scales, with the inherent assumption that the remaining squared wavelet coecients represent the energy associated only with the structure containing part of the turbulence signal. An ambiguity remains as how to choose the right threshold value, since various authors propose dierent values. Donoho and Johnstone10,11 and Donoho12 de-rived an algorithm to select a threshold value for near-optimal noise reduction in the presence of white noise. The application of the so derived threshold value was presented in the local method. However, since the non-structure component of atmospheric turbulent signals cannot be considered to be pure white noise due to the signi®cant slope of the corresponding non-structure energy spectrum21,22,42, the near-optimal behavior of the local method in these cases might not be guaranteed.
ACKNOWLEDGEMENTS
The authors are grateful to the three anonymous re-viewers for their valuable comments on a previous ver-sion of the manuscript.
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